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Dr. Bhairaba Kumar Majhi Category

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Code(Credit) : CUTM1519(3-0-3)

Course Objectives

• To point out that iterative processes and convergence of sequences occur in many areas of mathematics, and to develop a general context in metric spaces.
• To provide a basic course in analysis.
• To reinforce ideas of proof

Learning Outcomes

• Understand the Euclidean distance function on Rn and appreciate its properties, and state and use the Triangle and Reverse Triangle Inequalities for the Euclidean distance function on Rn
• Explain the definition of continuity for functions from Rn to Rm and determine whether a given function from Rn to Rm is continuous
• Explain the geometric meaning of each of the metric space properties  and be able to verify whether a given distance function is a metric
• Distinguish between open and closed balls in a metric space and be able to determine them for given metric spaces
• Define convergence for sequences in a metric space and determine whether a given sequence in a metric space converges
• State the definition of continuity of a function between two metric spaces.

Course Syllabus

Module-I (T-5hr P-0hr P-6hr)

Metric Spaces: Definition and examples, open and closed spheres,

Project-1: Write a report on applications of metric spaces.

Project-2: Show that the class denote a set of all bounded sequence of real numbers is a metric space.

Project-3: Show that the class of all sequences is a metric space.

Module-II (T-3hr P-0hr P-4hr)

Neighbourhoods, Interior points. Open set

Project-4: Write a report on relation between boundary points, limit points. And their applications.

Project-5(Flip class): Prove theorems on Neighbourhoods, Interior points

Module-III (T-5hr P-0hr P-6hr)

Closed set, boundary points, limit points and isolated points, closure of a set, dense sets.

Project-6: Write a report on difference between open set closed set and limit point , interior point with appropriate examples.

Project-7: Write the difference between boundary points, limit points and isolated points explain through examples.

Project-8 (Flip class): Prove theorems on closed set and dense sets.

Module-IV (T-4hr P-0hr P-6hr)

Separable metric spaces, sequences in metric space, convergent sequences, Cauchy sequences

Project-9: Write a report on applications of convergent sequences and Cauchy sequences.

Project-10: Every convergent sequence is bounded. Is the reverse true? Justify your answer with suitable example.

Project-11(Flip class): Prove theorems on convergent sequences and Cauchy sequences.

Module-V (T-4hr P-0hr P-4hr)

Complete metric space, subspace, Cantor’s theorem. Continuous functions,

Project- 12(Flip class): Prove theorems on Complete metric space, subspace.

Project-13(Flip class): : Prove that addition, subtraction, multiplication and division (with condition) of continuous functions are continuous.

Module-VI (T-4hr P-0hr P-6hr)

Uniform continuity, Connectedness, connected subsets of separated sets.

Project-14(Flip class): Prove theorems on Uniform continuity and Connectedness.

Project- 15: Write the difference between continuity and uniform continuity and explain through a proper example..

Project-16 (Flip class): Prove theorems on connected subsets of separated sets.

Module-VII (T-5hr P-0hr P-4hr)

Disconnected sets, contraction mappings, Banach fixed point theorem.

Project- 17(Flip class): Geometrical interpretation of connected and disconnected sets.

Project-18: Write some applications of Banach fixed point theorem.

Text Books:

1. K. Jain and K. Ahmad, Metric Spaces, Narosa Publishing House, New Delhi.
2. Methods of Real analysis-R.G. Goldberg.   Chapters: 4(4.2,and 4.3), 5,and  6 (up to 6.8)

Reference Books:

1. S.C. Malik and S. Arora – Mathematical Analysis (New Age International)
2. Principles of Mathematical Analysis – Walter Rudin, III Edition, McGRAW-Hill Book Co.

Session 2

Examples on Metric Spaces

Study Material: http://more examples

Session 3

Project-1: Write a report on applications of metric spaces.

Session 4

Theorems on Metric Spaces.

Study Material: Theorems on metric space

Session 5

Project-2: Show that the class denote a set of all bounded sequence of real numbers is a metric space.

Open spheres.

Closed spheres

Session 8

Project-3: Show that the class of all sequences is a metric space.

Session 9

Neighbourhood of a point

Session 10

Interior points and theorems

Session 11

Project-4: Write a report on relation between boundary points, limit points. And their applications

Session 12

Open set

Study material: open set

Session 13

Project-5(Flip class): Prove theorems on Neighbourhoods, Interior points

Session 14

closed set

study material: closed set and theorems

Session 15

limit points and isolated point

Session 16

Project-6: Write a report on difference between open set closed set and limit point , interior point with appropriate examples.

closure of a set

Session 18

Theorems on closurere of a set

dense set

Session 20

Project-7: Write the difference between boundary points, limit points and isolated points explain through examples

Session 21

Project-8 (Flip class): Prove theorems on closed set and dense sets.

Session 22

Separable metric spaces

Session 23

sequences in metric space

Session 24

convergent sequences

Session 25

Project-9: Write a report on applications of convergent sequences and Cauchy sequences

Session 26

Project-10: Every convergent sequence is bounded. Is the reverse true? Justify your answer with suitable example.

cauchy sequences

Session 28

Project-11(Flip class): Prove theorems on convergent sequences and Cauchy sequences.

Session 29

subspaces and theorems

Session 30

Complete metric space

Session 31

Project-12(Flip class): Prove theorems on Complete metric space, subspace.

Session 32

Continuous functions

Session 33

Project- 13: Prove that addition, subtraction, multiplication and division (with condition) of continuous functions are continuous.

Session 34

Theorems on Continuous functions

Session 35

Uniform continuity and Theorems

Connectedness

Session 37

Project-14(Flip class): Prove theorems on Uniform continuity and Connectedness.

Session 38

Theorems on Connectedness

Session 39

Project- 15: Write the difference between continuity and uniform continuity and explain through a proper example

Session 40

Theorems on Connected and separated sets

Session 41

Project-16 (Flip class): Prove theorems on connected subsets of separated sets.

Session 42

Disconnected sets

Session 43

Theorems on Disconnected sets

Session 44

Project- 17(Flip class): Geometrical interpretation of connected and disconnected sets.

Session 45

Banach fixed point theorem.

Session 46

contraction mapping

Session 47

contraction mapping and theorems

Session 48

Project-18: Write some applications of Banach fixed point theorem.

Case Studies 